Consequently, we deduce by stochastic Gronwall's inequality (see
e.g. [1]}) that, for \(0<\kappa ^{\prime }<\kappa <1\) ,
\(\begin{split}&\Big (\mathbb {E}\Big (\sup _{0\le s\le t}|Z_s^{(\delta )}|^{2\kappa ^{\prime }}\Big )\Big )^{1/\kappa ^{\prime }}\\&\le \Big (\frac{\kappa }{\kappa -\kappa ^{\prime }}\Big )^{1/\kappa ^{\prime }}\Big (\mathbb {E}\text{\rm {e}}^{\kappa A_t/(1-\kappa )}\Big )^{(1-\kappa )/\kappa }\times \int _0^t\Big \lbrace C_3\mathbb {E}|b(X^{(\delta )}_s)-b(X^{(\delta )}_{s_\delta })|^2\Big \rbrace \text{\rm {d}}s.\end{split}\) 
